Decision diagrams in machine learning: an empirical study on real-life credit-risk data.

Decision trees are a widely used knowledge representation in machine learning. However, one of their main drawbacks is the inherent replication of isomorphic subtrees, as a result of which the produced classifiers might become too large to be comprehensible by the human experts that have to validate them. Alternatively, decision diagrams, a generalization of decision trees taking on the form of a rooted, acyclic digraph instead of a tree, have occasionally been suggested as a potentially more compact representation. Their application in machine learning has nonetheless been criticized, because the theoretical size advantages of subgraph sharing did not always directly materialize in the relatively scarce reported experiments on real-world data. Therefore, in this paper, starting from a series of rule sets extracted from three real-life credit-scoring data sets, we will empirically assess to what extent decision diagrams are able to provide a compact visual description. Furthermore, we will investigate the practical impact of finding a good attribute ordering on the achieved size savings.Advantages; Classifiers; Credit scoring; Data; Decision; Decision diagrams; Decision trees; Empirical study; Knowledge; Learning; Real life; Representation; Size; Studies;

Discrete Function Representations Utilizing Decision Diagrams and Spectral Techniques

All discrete function representations become exponential in size in the worst case. Binary decision diagrams have become a common method of representing discrete functions in computer-aided design applications. For many functions, binary decision diagrams do provide compact representations. This work presents a way to represent large decision diagrams as multiple smaller partial binary decision diagrams. In the Boolean domain, each truth table entry consisting of a Boolean value only provides local information about a function at that point in the Boolean space. Partial binary decision diagrams thus result in the loss of information for a portion of the Boolean space. If the function were represented in the spectral domain however, each integer-valued coefficient would contain some global information about the function. This work also explores spectral representations of discrete functions, including the implementation of a method for transforming circuits from netlist representations directly into spectral decision diagrams

Advances in Functional Decomposition: Theory and Applications

Functional decomposition aims at finding efficient representations for Boolean functions. It is used in many applications, including multi-level logic synthesis, formal verification, and testing. This dissertation presents novel heuristic algorithms for functional decomposition. These algorithms take advantage of suitable representations of the Boolean functions in order to be efficient. The first two algorithms compute simple-disjoint and disjoint-support decompositions. They are based on representing the target function by a Reduced Ordered Binary Decision Diagram (BDD). Unlike other BDD-based algorithms, the presented ones can deal with larger target functions and produce more decompositions without requiring expensive manipulations of the representation, particularly BDD reordering. The third algorithm also finds disjoint-support decompositions, but it is based on a technique which integrates circuit graph analysis and BDD-based decomposition. The combination of the two approaches results in an algorithm which is more robust than a purely BDD-based one, and that improves both the quality of the results and the running time. The fourth algorithm uses circuit graph analysis to obtain non-disjoint decompositions. We show that the problem of computing non-disjoint decompositions can be reduced to the problem of computing multiple-vertex dominators. We also prove that multiple-vertex dominators can be found in polynomial time. This result is important because there is no known polynomial time algorithm for computing all non-disjoint decompositions of a Boolean function. The fifth algorithm provides an efficient means to decompose a function at the circuit graph level, by using information derived from a BDD representation. This is done without the expensive circuit re-synthesis normally associated with BDD-based decomposition approaches. Finally we present two publications that resulted from the many detours we have taken along the winding path of our research

Boolean Expression Diagrams

This paper presents a new data structure called Boolean Expression Diagrams (BEDs) for representing and manipulating Boolean functions. BEDs are a generalization of Binary Decision Diagrams (BDDs) which can represent any Boolean circuit in linear space and still maintain many of the desirable properties of BDDs. Two algorithms are described for transforming a BED into a reduced ordered BDD. One is a generalized version of the BDD apply-operator while the other can exploit the structural information of the Boolean expression. This ability is demonstrated by verifying that two di erent circuit implementations of a 16-bit multiplier implement the same Boolean function. Using BEDs, this veri cation problem is solved in less than a second, while using standard BDD techniques this problem is infeasible. Generally, BEDs are useful in applications, for example tautology checking, where the end-result as a reduced ordered BDD is small

Synthesis and Optimization of Reversible Circuits - A Survey

Reversible logic circuits have been historically motivated by theoretical research in low-power electronics as well as practical improvement of bit-manipulation transforms in cryptography and computer graphics. Recently, reversible circuits have attracted interest as components of quantum algorithms, as well as in photonic and nano-computing technologies where some switching devices offer no signal gain. Research in generating reversible logic distinguishes between circuit synthesis, post-synthesis optimization, and technology mapping. In this survey, we review algorithmic paradigms --- search-based, cycle-based, transformation-based, and BDD-based --- as well as specific algorithms for reversible synthesis, both exact and heuristic. We conclude the survey by outlining key open challenges in synthesis of reversible and quantum logic, as well as most common misconceptions.Comment: 34 pages, 15 figures, 2 table